If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α > β and P n = α n - β n for each positive integer n , then the value of P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 is equal to

Step-by-step solution

Given, x 2 + 5 2 x + 10 = 0 and P n = α n - β n Now P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 = P 17 P 20 + 5 2 P 19 P 18 P 19 + 5 2 P 18 = P 17 α 20 - β 20 + 5 2 α 19 - β 19 P 18 α 19 - β 19 + 5 2 α 18 - β 18 = P 17 α 19 α + 5 2 - β 19 β + 5 2 P 18 α 18 α + 5 2 - β 18 β + 5 2 Since α + 5 2 = - 10 / α               . . . 1 &   β + 5 2 = - 10 / β   &#16